The Hp Type Galerkin And Spectrum Collocation Method For Nonlinear Delay Differential Equations

In this paper,we investigate the h-p version time stepping methods for first order nonlinear delay differential equations.On the one hand,we propose an h-p version of the continuous Petrov-Galerkin method for the nonlinear delay differential equations with vanishing delays.We derive a-priori error estimates in the L2-,H1-and L∞-norms that are completely explicit with respect to the local time steps,the local polynomial degrees,and the local regularity of the exact solution.In particular,we show that the h-p version continuous Petrov-Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders achieves exponential rates of convergence for solutions with start-up singularities.On the other hand,we propose an h-p version Chebyshev-Gauss-Lobatto spectral collocation method for the nonlinear delay differential equations with vanishing delays.We design a fast algorithm with high accuracy and derive an h-p version error estimate in the H1-norm.The theoretical results are illustrated by some numerical examples.